PREFACE

What is Knowledge Bytes ?

Knowledge Bytes is a collection of riddles, interesting facts, mnemonics
and puzzles that will make your learning fun and engaging.
We want you to be delighted about studying. Knowledge Bytes helps you to
know more about the subject in a fun, motivating and educational way and
helps to implement what you learn in a creative way.

Benefits
Saves Time

Develops Learning Skills

Stimulates Interest

Leads to Increased Comprehension

EXPLORE
1. Real Numbers 1
2. Light-Reflection 9
3. Chemical Reactions and Equations 13
4. Life Processes- Nutrition and Respiration 18
5. Sources of Energy 24
6. Tenses 28
7. Quantitative Aptitude : Profit, Loss and Discount 31
 Real Numbers

Crossword
Across 1 2
3
1. Every real number can be represented 4
5 6
by a _____ point on the number line. [6] 7 8
9
8. Product of two rational numbers is
_____. [8] 10

9. 11
Euclid’s division lemma can be extended
for all whole numbers except _____. [4]

10. HCF of 14 and 7 is _____. [5]

11. There are _____ rational numbers between any two rational numbers. [8]

Down
2. If the LCM and HCF of two numbers are equal, then the numbers must be
_____. [5]

3. 3 + 5 is a/an _____ number. [10]

4. Decimal expansion of an irrational number is _____ and non-repeating. [14]

5. Product of two numbers is equal to the _____ of their HCF and LCM. [7]

6. Dividend = Divisor x _____ + Remainder.[8]

7. The decimal number in which all the figures after the decimal point are
1
repeated is called _____ decimal. [9]
Number System
The number system or the numeral system is the system of naming or
representing numbers. There are various types of number systems in
maths like binary, decimal, etc.

What is Number System

in Maths?

A number system is defined as a writing system to

express numbers. It is the mathematical notation for
representing numbers of a given set by using digits or other
symbols in a consistent manner.

The value of any digit in a number can be determined by:

The digit
Its position in the number
The base of the number system

What is Number ?

A number is a mathematical value used for

counting or measuring or labeling objects.
Numbers are used to perform arithmetic
calculations. Numbers are classified as natural
numbers, whole numbers, real numbers, rational
and irrational numbers, etc.
2
 Types of Number System

The four most common number system types are:

1. Decimal number system (Base-10)

2. Binary number system (Base-2)

3. Octal number system (Base-8)

4. Hexadecimal number system (Base-16)

Decimal Number System (Base-10)

Decimal number system has base 10 because it uses ten digits from 0 to 9.
For example, the decimal number 1457 consists of the digit 7 in the units
place, 5 in the tens place, 4 in the hundreds place, and 1 in the thousands
place whose value can be written as
= (1 × 103) + (4 × 102) + (5 × 101) + (7 × 100)

= (1 × 1000) + (4 × 100) + (5 × 10) + (7 × 1)

= 1000 + 400 + 50 + 7

= 1457

Binary number system (Base-2)

The Base-2 number system is also known as the Binary number system
wherein, only two binary digits exist, i.e., 0 and 1. The figures described under
this system are known as binary numbers which are the combination of 0
and 1. For example, 110101 is a binary number.
We can convert any system into binary and vice versa.
3
 Write (13)10 as a binary number.

Example
Sol:- 2 13
2 6 1
2 3 0
1 1
∴ (13)10 = (1101)2

Octal Number System (Base-8)

In the octal number system, the base is 8 as it uses numbers from 0 to 7 to

represent numbers. Octal numbers are commonly used in computer
applications.

Convert (215)8 into decimal.

Sol:- (215)8 = 2 × 82 + 1 × 81 + 5 × 80
Example

= 2 × 64 + 1 × 8 + 5 × 1

= 128 + 8 + 5

= 141

Hexadecimal Number System (Base-16)

In the hexadecimal number system, numbers are written or represented with

base 16. In this system, the numbers are represented by the digits 0 to 9 and
alphabets A to F. Any number in hexadecimal number system can be
represented by using these 16 alphanumeric characters.

Hexadecimal 0 1 2 3 4 5 6 7 8 9 A B C D E F
Decimal 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
4
 Mysteries of Numbers
Mystery of Number 9
It is said to be a significant part of Indian Psyche.
1. The ‘Fruit of the Spirit' comprises nine graces:" love, joy, peace,
patience, kindness, goodness, faithfulness, gentleness, and self-
control.

2. Nine being a significant part of the Indian

psyche from time immemorial, the creators
of our classical arts also found nine
emotions or artistic moods that form a part
of our expressions in life.

3. THE NUMBER 18 (8 + 1 = 9) : Hindu epic Mahabharata is associated

with number 18 in several ways. The Mahabharata is divided into 18
books (parvas) and the Bhagavad Gita also has 18 chapters.
Mystery of Number 3

1. Almost everything in life can (and is) separated into a beginning, a

middle and an end.

2. Time is measured through past, present,

and future.

3. The dimensions of the space we inhabit are

measured through width, length and height.

4. The Goddess herself has three expressions: Virgin, mother and crone.

5. In astrology, the planet Jupiter vibrates to the frequency of the three.

6. The third sign of the Zodiac is Gemini (where the Sun sails from May
5
21 – June 21).
Mystery of Number 7

1. Spiritually, the number 7 is sacred and carries ‘the energy of the

mystics’.

2. The rainbow has seven colours.

3. There are seven continents on planet Earth.

4. There are seven seas on planet earth.

5. Every week has seven days.

6. The Lotus flower on Buddha’s pedestal has 7 petals.

7. There were 7 great ancient sages called as Saptarishi.

Convert (1056)16 to octal number.

Sol:- Given, 105616 is an hexadecimal number.

First we need to convert the given hexadecimal number into decimal
number
(1056)16

= 1 × 163 + 0 × 162 + 5 × 161 + 6 × 160

= 4096 + 0 + 80 + 6

= (4182)10

6
 Now we will convert this decimal number to required octal number by
repetitively dividing by 8.

8 4182 Remainder
8 522 6
8 65 2
8 8 1
8 1 0
0 1

Therefore, taking the value of remainder from bottom to top, we get;

(4182)10 = (10126)8

Therefore,
(1056)16 = (10126)8

Convert (1001001100)2 to decimal

number.

Sol:- (1001001100)2

= 1 × 29 + 0 × 28 + 0 × 27 + 1 × 26 + 0 × 25 + 0 × 24 + 1 × 23 + 1 × 22 + 0 × 21
+ 0 × 20
= 512 + 64 + 8 + 4

= (588)10

7
Answer (Crossword)

2
1 U N I Q U E 3
4 Q I
5 N U 6 R
7 P O A Q R
R 8 R A T I O N A L U A
9 Z E R O T O T
P D E T I
E U R I O
A C M 10 S E V E N
T T I N A
I 11 I N F I N I T E L
N A
G T
I
N
G

8
 Light-Reflection

A Map of The City in The Sky - Light Pillars

On Jan 13, 2016, by Mia Heikkilä in Eura, Finland shared a
photo of reflection of city into the sky. This splendid
phenomenon happens due to light pillars which contains the
ice crystal suspended in the sky. These ice crystals which has
reflecting surface act as a group of tiny mirrors.
Mia Heikkilä

Light pillars : Light pillars are optical

illusions caused by ice crystals suspended
in the air. These crystals can take on a
variety of shapes as water freezes, but a
common one is a flat, hexagonal crystal. They’re heavier than air, so they fall,
but if they’re the right size (bigger than about 20 microns across, 1/5th the
width of a typical human hair), they fall slowly and stay oriented flat, parallel
to the ground.
9
 Experiment to Bend the Light
Materials required – Empty bottle, water, thin metal pin, laser.

1. Poke a hole into the middle of a water bottle with a metal.

2. Fill the bottle with water.

3. Make sure the hole is in the middle or just below the middle of the bottle
so the water flows well.
4. Cover the hole with a piece of masking tape so it doesn’t leak
immediately when you add water.
5. Fill it completely to the top and leave the lid off. The pan will catch the
water as it drains out when you remove the tape.
6. Aim the beam of a laser pointer through the bottle at the hole.

w i l l see Water Air

you real.
And
n = 1.3 n = 1.00
g i n
ndin
Normal line
b e
the
Laser light

Angle of
incidence

Laser beam

Water
a ir

10
 1. Light is a form of energy which causes sensation of
Facts of Light
vision. It travels in straight line path.
2. If incident light after interacting with a boundary separating two media
comes back into the same medium, this phenomenon is called reflection.
3. Images can be either real or virtual.

4. Real image is formed by the actual intersection of reflected rays and it can
be obtained on a screen.
5. Virtual image is formed by the apparent meeting of reflected rays from a
mirror, when produced backwards. It cannot be obtained on a screen.
6. Laws of reflection : There are two laws of reflection.
(i) The incident ray, the reflected ray and the normal at the point of
incidence all lie in the same plane.
(ii) The angle of incidence is equal to the angle of reflection i.e., ∠i = ∠r.

7. Spherical mirror : There are two types of spherical mirrors, viz concave
mirror and convex mirror.
Concave mirror : A spherical mirror whose reflecting surface is curved
inwards, facing towards the centre of the sphere.
Convex mirror : A spherical mirror whose reflecting surface is curved
outwards opposite to centre of sphere.
8. Focal length of a mirror : It is the distance between the pole and the focus
of a spherical mirror. It is half of its radius of curvature. It is positive for a
convex mirror and negative for a concave mirror.

9. Mirror formula, 1 = 1 + 1 , gives the relationship between the object

f u v
distance (u), image distance (v) and the focal length (f) of a spherical

mirror.
11
 Quiz Time ?
1. Focal length of plane mirror is

A Infinity B Zero

C Negative D None of these

2. Image formed by plane mirror is

A Real and erect B Real and invereted

C Virtual and erect D Virtual and inverted

3. A concave mirror gives real, inverted and same size image if the object is
placed

A At F B At infinity

C At C D Beyond C

4. Shining of diamond is due to

A Reflection of light B Refraction of light

C Total internal reflection D Dispersion of light

5. In optics, an object which has higher refractive index is called

A Optically rarer B Optically denser

C Optical density D Refractive index

1. A 2. C 3. C 4. C
Answer

5. B 12
 Chemical Reactions
and Equations

Crossword 1

Across 2

1. Chemical species which undergo the 3

chemical change. [8] 4 5
2. Gas which is used for packaging of 6 7
freshly prepared chips to avoid
8
rancidity. [8]
4. Conversion of hydrogen peroxide into
water and oxygen gas in presence of
sunlight. [10]
6. During the reaction of lead nitrate with
potassium iodide ___________ colored
precipitate of lead iodide is formed. [6]
8. Conversion of solid CO2 to gaseous form is an example of ______ change. [8]
Down
1. In the open atmosphere oily food starts giving bad odour, it is called ______ [9]
3. Substance which does not take part in the chemical reaction but enhances
the rate of a chemical reaction. [8]
5. Combination reactions are generally ______ in nature. [10]
7. Bio-catalyst is also known as ______. [6]
8. Chemical reaction in which insoluble solid is formed. [13]
13
The Lead Iodide ‘Golden Rain’ Reaction

The Experiment

+ MIX

KI Pb(NO3)2 Pbl2 & KNO3

The lead iodide is more soluble in warm water than in cold. It dissolves
when heated, reappearing as ‘golden rain’ on cooling.

HEAT

Pbl2 & KNO3 Pbl2 & KNO3

COOL

Pbl2 & KNO3

14
The Explanation

2Kl + Pb(NO3)2 Pbl2 + 2KNO3

When the two solutions of potassium iodide (Kl) and lead nitrate (Pb(NO3)2)
react, they produce soluble potassium nitrate (KNO3) and insoluble lead
iodide (Pbl2). This is visible as a yellow precipitate in the solution.
The lead iodide produced dissolves in the solution if it is heated, causing
the yellow precipitate to disappear and leaving a colourless solution. If this
solution is allowed to cool slowly, crystals of lead iodide begin to form,
causing the glistening ‘golden rain’ effect.

Lead salts are toxic and can

cause lead poisoning so any
skin contact should be

of PbI2? ?
Yellow precipitate
? ?
Yellow crystals
of PbI ? ?
avoided. They are also harmful
to the environment.
? ? ? ? ? ? ?
2

? ? ? ?
? ? Onions
? ? reactions.
Why Do Onions Make You Cry ?
It involves chemical
? out ?enzymes ? ? ?
? flow
?
which ?converts?amino acids
? present ? ?
? in ?onion into ?sulphenic ? ?
? ? ? itself?into propanethial
? ? S-oxide, ? an ? ? ?
acid when their peels are removed. Sulfenic acid being
unstable rearranges
? ? gas. ?Propanethial
irritating ? ? is?a lachrymatory
S-oxide ? ? agent,? ? ?
? ? ? acid ?when it touches
? ? the water ? layer ?that?covers and?
meaning that it generates tears when it touches the eye. Propanethial S-oxide
turns into sulphuric
? But? just? like onions,
? ? your eyeballs.
protects ? your eyes ? with
? ?are also equipped ? a?
? ? agent,?they?generate?tears to flush
? a?lachrymatory ? ?it out. ?
defense mechanism meant to protect them from harm. When the nerves in
each eye detect
? ? ? ? ? ? ? ? ? ? ?
15

? ? ? ?
 Quiz Time ?
1. Ratio of the gases collected at anode and cathode during the electrolysis of
water is
A 1:2 B 2:3

C 1:1 D 1:3

2. Which of the following compounds does not undergo photolysis?

A Silver chloride B Silver bromide

C Calcium carbonate D Hydrogen peroxide

3. aMnO2 + bHCl → cMnCl2 + dH2O + eCl2

In the given reaction a, b, c, d, e are
A 1, 4, 1, 2, 2 B 2, 4, 2, 2, 1

C 1, 4, 1, 2, 1 D 1, 1, 1, 2, 1

4. N2 + O2 → 2NO is an

A Exothermic combination reaction

B Endothermic combination reaction
C Exothermic decomposition reaction
D Endothermic decomposition reaction

5. Identify the odd one with respect to type of decomposition reaction.

A 2FeSO4 → Fe2O3 + SO2 + SO3 B CaCO3 → CaO + CO2

C 2H2O2 → 2H2O + O2 D 2Pb(NO3)2 → 2PbO + 4NO2 + O2

16
Answer (Crossword)

1 R E A C T A N T
A
2 N I T R O G E N
3 C
C I
A D 5
4 P H O T O L Y S I S E
A T 7 X
L 6 Y E L L O W
Y N T
8 P H Y S I C A L Z H
R T Y E
E M R
C E M
I I
P C
I
T
A
T
I
O
N

Answer (Quiz Time)

1. A 2. C 3. C 4. B 5. C

17
 Life Processes-
Nutrition and Respiration

1. Chewing Over The Conundrum!

a
This gland might claim to be the largest but it doesn’t secrete
any protein.

I look like a million fingers on the inside of a tube & can quickly

b help you to uptake the nutrients present in the food.

I am a versatile gland that helps your intestines. I have a duct &

c yet you can call me endocrine.

I pour the saliva inside your mouth, but hide below your ear. You

d might see a swelling when a virus grows in here.

18
e
I may not be able to digest anything but I can pass your food. I
may not have bones but I have my rhythmic moves.

I am a J-shaped bag full of acid. I churn your food for hours at

f a stretch & may sometimes make you belch.

g
I am not an organism but I have some teeth. I ingest & partially
digest your pizza with extra cheese!

I am the largest part of large intestine. I go up, side & down &

h slide your food right out on time.

I prevent any slipping of food in the lungs. I am a small flat part

i at the back of the tongue.

Unscramble all the letters in the circle to make a word.

Hint: It is the process that connects them all.

19
 Baffling Breaths

a. Approx. 70% wastes of the body are eliminated in the form of gases.

b. The body inhales because of high carbon dioxide levels in the blood rather
than low levels of oxygen. This is regulated by the medulla oblongata in
the brain.

c. A surfactant, produced within the alveoli, reduces the surface tension

present & prevents it from collapsing when we exhale.

d. Lungs are the only body organs that can float on water, because of the air
in alveoli. In fact, doctors employ the "lung float test" during autopsies to
determine if a baby was born dead or alive. If they don't float, it was a still
birth, but if it does, the baby was born alive.

e. Many people in the world are living with just one lung. Although it might
alter stamina, but survival is fairly possible.

f. For balancing the breath between the nostrils, our body automatically
changes sides for every 30 min.

g. An average human adult exhales up to 17.5 ml of water/hr.

h. When you exercise, your fat burns & converts into CO2, eliminated out as
gas. But you cannot just breathe more to lose weight, you have to exercise
more & eat less.

i. When we breathe in, oxygen is carried by the iron in the haemoglobin of our
RBCs. Iron is only formed in the space due to supernova collisions.

20
 Emphysema
The word ‘Emphysema’ means ‘full of air’ or ‘inflation’. It is a chronic disorder.
Causes : The major cause is excessive cigarette smoking. Others may
include inhalation of smoke or toxic substances over a period of time.
Symptoms : The walls of alveoli are damaged due to excessive
smoking, loss of elasticity of walls of bronchioles and
alveoli. Due to this, the surface area for exchange of
gases is reduced. Alveolar sacs remain filled with air
even after expiration. The lungs remain inflated as
exhalation becomes difficult. This disease may be
prevented by avoiding cigarette smoking.

Terms Associated with Breathing

Eupnea (i) Normal breathing

Hypopnea (iii) Slow breathing

Hyperpnea (iii) Rapid breathing

Apnea (iv) No breathing

Dyspnea (v) Painful breathing

21
2. Label the following parts from A to J in the given diagram of human
digestive system.

A B
Mouth C
Submaxillary
and sublingual glands
Oesophagus
Liver

F
Stomach
G D
H Jejunum
Ascending colon E
I Sigmoid colon
J Rectum
Anus

3. Which of the following is an incorrect match regarding reaction, its location

of occurrence and the enzyme involved?

Reaction Location Enzyme

pH 1.8
(i) Protein Proteoses + peptones Stomach Pepsin

(ii) Procarboxy Carboxypeptidase Duodenum Trypsin

peptidase
(iii) Fats Diglycerides + Monoglyceride Stomach Steapsin
(iv) Dipeptides Amino acid Duodenum Dipeptidase

1 (i) 2 (ii)

3 (iii) 4 (iv)
22
Answers
1. a. L I V E R f. S T O M A C H

b. V I L L I g. M O U T H

c. P A N C R E A S h. C O L O N

d. P A R O T I D i. E P I G L O T T I S

e. O E S O P H A G U S D I G E S T I O N

2. A Oral cavity F Gall bladder

B Parotid gland G Duodenum

C Pharynx H Transverse colon

D Pancreas I Ileum

E Descending colon J Caecum

3. (iii)

23
 Sources of Energy

Non-conventional Sources of Energy and Their Location

Nuclear or Atomic Energy

Uranium and Thorium -
Jharkhand and the Aravalli
ranges of Rajasthan.

Thorium - Monazite sands

of Kerala.

India’s thorium is mostly located in a

contiguous belt of eastern coastal states
as placer sands.
2016 Monazite reserve estimates.

Andhra Pradesh (31%)

Tamil Nadu (21%) and Odisha (20%)
Kerala (16%) and West Bengal (10%)
Jharkhand (2%)
24
 DADAR
GUJRAT
Solar Energy

JAMMU
& KASHMIR
Tropical Region of India

LADAKH

Ka
rn
at
ak
Te a 6095.55
l an
ga
na
Ra 3592.09
ja
st
An ha
dh n 3226.79
r a
Pr
ad
e sh
Ta 3085.68
m
il
Na
du
2575.22
Gu
M ja
ad rat
hy 2440.13
a
Pr
ad
es
M h 1840.16
ah
ar
as
ht
Ut ra 1633.54
ta
rP
ra
de
sh 960.10 Top 10 State in Solar Capacity (in MW, till March 2019)
Pu
nj
NAGALAND

ab
905.62

25
Wind Power
Tamil Nadu, Andhra Pradesh, Karnataka, Gujarat, Kerala, Maharashtra,
and Jaisalmer (Rajasthan).

LADAKH

JAMMU
& KASHMIR

NAGALAND

GUJRAT

TELANGANA
SH
ADE
PR
A
DHR
AN

26
Tidal Energy
Gulf of Khambhat, Gulf of Kuchchh and Gangetic Delta in Sunderban
Regions
Gulf of Cambay/
Khambat
Potential: 7000 MW

Gulf of Kutch
Potential: 1200 MW 10-12 MW
8-10 MW
6-8 MW
Sunderbans Delta 4-6 MW
2-4 MW
1-2 MW
Potential: 100 MW <1 MW

Geo Thermal Energy

Parvati valley near
Manikarn in
Himachal Pradesh.
Puga Valley, Ladakh.

27
 Tenses

Choose the correct tense form to fill in the blank.

1. He did not ______ part in the debate.

(A) took (B) taken
(C) take (D) takes

2. He _____ not taken anything till then.

(A) have (B) has
(C) had (D) did

3. Has he not ______ to school for many days?

(A) coming (B) came
(C) come (D) been coming

4. Her train ______ at 8.35 a.m. tomorrow.

(A) arrives (B) is arriving
(C) will arrive (D) arrive
28
5. The light went out while I ______.
(A) reading (B) were reading
(C) am reading (D) was reading

6. The father with his two sons _______ to witness the match.
(A) is going (B) was going
(C) are going (D) were going

7. Scoring well in Mathematics ______ in getting first division.

(A) help (B) will be help
(C) helps (D) helped

8. She _____ awake for a while now to finish her project at the earliest.
(A) was (B) had been
(C) has been (D) was being

9. Had the bell ______ before you reached home?

(A) ring (B) rang
(C) rung (D) has rung

29
10. One hardly ______ what to do.
(A) knew (B) know
(C) knows (D) known

Answers 1. (C) 2. (C) 3. (D) 4. (A) 5. (D)

6. (A) 7. (C) 8. (C) 9. (C) 10. (C)

30
 Quantitative Aptitude :
Profit, Loss and Discount

1. After selling 70 articles a man earns a profit of cost price of 9 articles. Find
profit percentage.

1
A 12% B 12 %
8
6 5
C 12 % D 12 %
7 7

2. Cost price of 10 articles is equal to selling price of 8 articles. What is the

profit percentage?

A 20% B 25%

C 22% D 26%

3. Cost price of 12 articles is equal to selling price of 9 articles while discount

on 10 articles is equal to profit on 8 articles. What is the difference between
profit % and discount %?

A 50 % B 40%
3

C 52% D 47%

31
4. A dishonest shopkeeper promise to sell his goods at its cost price. But he
uses 950 grams weight instead of 1 kg. Find profit percentage.

5 5
A 5 % B 9 %
19 4
5 1
C 4 % D 4 %
9 4

5. A shopkeeper allow 20% discount on marked price and earn 25% profit. If he
gets `90 as profit. What is the amount of the discount?

A `110.5 B `114.5

C `101.5 D `112.5

Answer

1. (C) Cost price of 1 article = ` 1

Profit = ` 9

CP of 70 articles = ` 70
9 6
Profit % = × 100 = 12 %
70 7

2. (B) CP × 10 = 8 × SP
CP 4 SP – CP 5–4
= ⇒ × 100 = × 100 = 25% profit
SP 5 CP 4

3. (A) CP → Cost price, D → Discount, P → Profit, SP → Selling price

SP 4
CP × 12 = SP × 9 ⇒ =
CP 3
32
 SP – CP 4–3 100
⇒ Profit % = × 100 = × 100 =
CP 3 3
D 4
D × 10 = P × 8 ⇒ =
P 5
MP → Marked price

CP SP MP
3 4 x Profit = 1
15 20 5x Profit = 5

24
5x – 20 = 4 ⇒ x =
5
D 4 100 50
D% = × 100 = × 100 = =
MP 24/5 × 5 6 3

50 100 50
D% – P% = – =–
3 3 3
50
So, required difference = %
3

4. (A) 1000 gm

CP = 950 SP = 1000

+ 50

Profit = 50

50 5
Profit % = × 100 = 5 %
950 19
33
5. (D) CP SP MP D
1
4×5 5×5 20% =
5
MP
4×4 5×4 SP = 4
P
1
= 16 20 25 25% =
4
CP

90
Profit = 4 units, so 4 units → 90 ⇒ 1 →
4

Discount = 25 – 20 = 5

90 450 1
=5× = = 112
4 4 2

34
Happy Birthday
Stephen Hawking

The greatest enemy of

knowledge is not
IGNORANCE, it is the
ILLUSION of knowledge.

Stephen Hawking was born in Oxford, England

on January 8, 1942. He grew up in a highly
educated family. Both of his parents had
attended Oxford University and his father, Frank,
was a medical researcher.

Hawking’s contributions to physics earned him many

exceptional honours. In 1974 the Royal Society elected
him one of its youngest fellows. He became professor of
gravitational physics at Cambridge in 1977, and in 1979
he was appointed to Cambridge’s Lucasian professorship
of mathematics, a post once held by Isaac Newton.